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Theorem elin2 3532
Description: Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Hypothesis
Ref Expression
elin2.x  |-  X  =  ( B  i^i  C
)
Assertion
Ref Expression
elin2  |-  ( A  e.  X  <->  ( A  e.  B  /\  A  e.  C ) )

Proof of Theorem elin2
StepHypRef Expression
1 elin2.x . . 3  |-  X  =  ( B  i^i  C
)
21eleq2i 2501 . 2  |-  ( A  e.  X  <->  A  e.  ( B  i^i  C ) )
3 elin 3531 . 2  |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
42, 3bitri 242 1  |-  ( A  e.  X  <->  ( A  e.  B  /\  A  e.  C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    i^i cin 3320
This theorem is referenced by:  elin3  3533  fnres  5562  funfvima  5974  fnwelem  6462  fz1isolem  11711  isabl  15417  isfld  15845  2idlcpbl  16306  divs1  16307  divsrhm  16309  isidom  16365  lmres  17365  isnvc  18731  ishl  19317  ply1pid  20103  rplogsum  21222  isphg  22319  ishlo  22390  hhsscms  22780  mayete3i  23231  elpredim  25452  elpred  25453  elpredg  25454  sltres  25620  nofulllem5  25662  caures  26467  iscrngo  26608  fldcrng  26615  isdmn  26665  isolat  30011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959  df-in 3328
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